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Proof step 6 from proof gc-heap-allocate-h (previous step)

lpt[n] = mklp(m0, m1, LP_FREE, m2), lpt[m3] = mklp(m, m4, lpf, m5), m3 < # lpt, n < # lpt, ∀ lpt, n, m0, m1, m2, lp, nx. lpt[n] = mklp(m0, m1, LP_FREE, m2) ∧ n < # lpt ∧ m3 ≤ # lpt → gc-heap(lpt[n, lp], nx + n ', m3) = gc-heap(lpt, nx, m3) ⊦ gc-heap(lpt[n, lp], nx + n ', m3 + 1) = gc-heap(lpt, nx, m3) ∪ (¬ m3 ∈ nx ∧ lpf = LP_GROUP_NODES ⊃ ∅ ++ key-index(m, m3);∅)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ n < # lpt → gc-heap(lpt, nx, n + 1) = gc-heap(lpt, nx, n) ∪ (¬ n ∈ nx ∧ lpt[n].flags = LP_GROUP_NODES ⊃ {key-index(lpt[n].ref-size, n)} ;∅)
Proof step 8:
lpt[m3] = mklp(m, m4, lpf, m5), lpt[n] = mklp(m0, m1, LP_FREE, m2), n < # lpt, m3 < # lpt, m3 < # lpt[n, lp], ∀ lpt, n, m0, m1, m2, lp, nx. lpt[n] = mklp(m0, m1, LP_FREE, m2) ∧ n < # lpt ∧ m3 ≤ # lpt → gc-heap(lpt[n, lp], nx + n ', m3) = gc-heap(lpt, nx, m3) ⊦ gc-heap(lpt[n, lp], nx + n ', m3) ∪ (¬ m3 ∈ nx + n ' ∧ lpt[n, lp][m3].flags = LP_GROUP_NODES ⊃ {key-index(lpt[n, lp][m3].ref-size, m3)} ;∅) = gc-heap(lpt, nx, m3) ∪ (¬ m3 ∈ nx ∧ lpf = LP_GROUP_NODES ⊃ ∅ ++ key-index(m, m3);∅)

Rule argument

spec: <toplevel> name: gc-heap-rec seq: ⊦ n < # lpt → gc-heap(lpt, nx, n + 1) = gc-heap(lpt, nx, n) ∪ (¬ n ∈ nx ∧ lpt[n].flags = LP_GROUP_NODES ⊃ {key-index(lpt[n].ref-size, n)} ;∅ ) subst: [n, nx, lpt] → [m3, nx + n ', lpt[n, lp]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # lpt[n, lp] = # lpt; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: m3, nx + n ', lpt[n, lp]